Repeater design using round robin transmitters to increase duty cycle?

It is likely that you can operate the TK-890H at 100% duty with better cooling. This would be a lot less work than the scheme you propose.

The Kenwood TK-890H has no fan - a good thing for mobile use, it's quiet and it will stay clean and last longer. But to keep it compact, the heatsink is much too small for continuous transmission.

My reasoning: The radio must however be able to tolerate a single long over without burning out or reducing power - at least 20 seconds of continuous talking. There is nothing in the electronics that can have such a long thermal time constant - the components on the PCB will heat up to full temperature in a few seconds, and the power transistors will be fully warmed up and dissipating into the heatsink. The only thing that is "slow" enough to be OK at 20 seconds, but not OK at 60 seconds, is the big heatsink itself. With a small heatsink, they're relying on the aluminium to sink all the energy from the transmission period, and dissipate it over a longer period. You can see that in the design too - it's heavy on metal, light on fins, so it can store more heat.

By adding a fan, you can dramatically improve the cooling effectiveness of a heatsink. Scratching on Digikey you'll find examples of heatsink thermal performance changing from 0.6 deg/watt (100 watts = 60 degrees rise, too hot) to 0.15 deg C/watt (100 watts = 15 degrees rise, OK) with forced air.

So I would say first test one radio with some fans blowing on it, through the heatsink in the best way you can think of. An array of fans would be better for tolerance of failure. I recommend small 10000+ RPM fans typically used in 1U servers as these move a large volume of air. They could be thermostatically controlled, or just use a timer, for the time it's transmitting plus 10 minutes. It might help to remove the case of the radio and have some air blowing over the PCB.

Finally, of course, the fans don't help if the air stays in a sealed box - they just make it worse by increasing the thermal load in the box. You'll need to have some sort of vent to the outside, with a filter to keep things clean inside.

Sure, I suppose it's possible. Any sort of "voting" seems overcomplicated to me. Simply have a timer that switches to the next radio every 30 seconds or so. If you have five of these radios and switch between them periodically, no one will have a duty cycle above 20%. If you have less than 5 radios then reduce power.

The complicated part I suppose is sequencing the switching. You need to:

1. Stop transmitting on radio A
2. Disconnect radio A
3. Connect radio B
4. Start transmitting on radio B

You need a little delay between each of these steps to wait for relays to react, etc. If you connect radio B while radio A is still transmitting, radio B's receiver will probably be blown out. If you transmit on radio B while radio A is transmitting, similarly bad things are likely to happen.

Of course there will be a glitch in the transmission every time a switch happens, but probably you can make it fast enough that it's not a serious problem. And I doubt you'll find any off the shelf thing to do this, so you'll have to build it yourself. The sequencing could be done by a simple microcontroller.

Alternatively, you could open the radios and extract their power amplifiers, operate each at a reduced power that could be sustained at 100% duty cycle, then combine their output power.